Geodesic
On the generalized Oberwolfach problem
Ars Mathematica Contemporanea, Tome 17 (2019) no. 1, pp. 67-78.

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The generalized Oberwolfach problem OPt(2w + 1; N1, N2, …, Nt; α1, α2, …, αt) asks for a factorization of K2w + 1 into αi CNi-factors (where a CNi-factor of K2w + 1 is a spanning subgraph whose components are cycles of length Ni ≥ 3) for i = 1, 2, …, t. Necessarily, N = lcm(N1, N2, …, Nt) is a divisor of 2w + 1 and w = Σ ti = 1 αi.For t = 1 we have the classic Oberwolfach problem. For t = 2 this is the well-studied Hamilton-Waterloo problem, whereas for t ≥ 3 very little is known.In this paper, we show, among other things, that the above necessary conditions are sufficient whenever 2w + 1 ≥ (t + 1)N, αi > 1 for every i ∈ {1, 2, …, t}, and gcd (N1, N2, …, Nt) > 1. We also provide sufficient conditions for the solvability of the generalized Oberwolfach problem over an arbitrary graph and, in particular, the complete equipartite graph.
DOI : 10.26493/1855-3974.1426.212
Keywords: 2-factorizations, resolvable cycle decompositions, cycle systems, (generalized) Oberwolfach problem, Hamilton-Waterloo problem 
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Andrea C. Burgess; Peter Danziger; Tommaso Traetta. On the generalized Oberwolfach problem. Ars Mathematica Contemporanea, Tome 17 (2019) no. 1, pp. 67-78. doi : 10.26493/1855-3974.1426.212. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1426.212/

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